I think you need to review the statement of the mean value theorem and the theory in general because it your last post indicates you don't understand it well. The MVT works for FUNCTIONS not simply equations.

The problem REQUIRES you to CONSTRUCT a polynomial function based on the constants a_0,..,a_n - it is here that you are required to be somewhat clever (if you've ever differentiated or integrated before the proper function you need to create it obvious) And from there use the two equations given to show that the function vanishes at 1 and at 0, and the rest is an application of the mean value theorem.

i started it three times with the mean value theorem (roll's, general mean value theorem,& integral mean value theorem), how ever, it didn't work so if you were in my case what will be your way of solving this question!!!